Goldstine theorem

In functional analysis, a branch of mathematics, the Goldstine theorem, named after Herman Goldstine, is stated as follows:

Goldstine Theorem. Let

X

be a Banach space, then the image of the closed unit ball

B \subset X

under the canonical embedding into the closed unit ball

B''

of the bidual space

X''

is weak*-dense.

The conclusion of the theorem is not true for the norm topology, which can be seen by considering the Banach space of real sequences that converge to zero, $c 0$ , and its bi-dual space $ℓ \infty$ .

Proof

Given $x'' \in B''$ , an $n$ -tuple $(φ 1, ..., φ n)$ of linearly independent elements of $X'$ and a $δ > 0$ we shall find $x$ in $(1 + δ) B$ such that $φ i (x) = x''(φ i)$ for $1 \leq i \leq n$ .

If the requirement $|| x || \leq 1 + δ$ is dropped, the existence of such an $x$ follows from the surjectivity of

{\begin{cases}\Phi :X\to {\mathbf {C}}^{{n}},\\x\mapsto \left(\varphi _{1}(x),\cdots ,\varphi _{n}(x)\right)\end{cases}}

Now let

Y:=\bigcap _{i}\ker \varphi _{i}=\ker \Phi .

Every element of $(x + Y) \cap (1 + δ) B$ has the required property, so that it suffices to show that the latter set is not empty.

Assume that it is empty. Then $dist(x, Y) \geq 1 + δ$ and by the Hahn-Banach theorem there exists a linear form $φ \in X'$ such that $φ | Y = 0, φ (x) \geq 1 + δ$ and $|| φ || X' = 1$ . Then $φ \in span{φ 1, ..., φ n},$ and therefore

1+\delta \leq \varphi (x)=x''(\varphi )\leq \|\varphi \|_{{X'}}\left\|x''\right\|_{{X''}}\leq 1,

which is a contradiction.

Functional analysis

Set / subset types	Absolutely convex Absorbing Balanced Bounded Convex Radial Star-shaped Symmetric Linear cone (subset) Convex cone (subset)

TVS types	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Webbed Topological tensor product (of Hilbert spaces)

Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence

Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Set operations	Algebraic interior (core) Interior Minkowski addition Polar

Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition

Theorems	Banach–Alaoglu Banach–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function

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Goldstine theorem

Proof

See also